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8/9/2019 Portfolio VaR Estimation_Final1leen
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PORTFOLIOVAR
ESTIMATION
Husam Sawalha Muna GhoshehLeen Bargouth Flora Mansour
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INTRODUCTION
A portfolio was built by choosing the following stocks
from Amman stock exchange :
Jordan Ahli Bank. The Jordan Cement Factories. Jordan Poultry Processing & Marketing. Jordan Diary.
The historical data of those stocks was collected for themost recent 501 days
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INTRODUCTION
Our Companies Value of investment
AHLI 4000
JO CEMENT 3000
JO POULTARY 2000
JO DAIRY 1000
Total 10,000
J.D 10,000 was invested in this portfolio allocated
as follow :
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INTRODUCTION
Portfolio VaR was estimated according to the
following approaches:Standard Approach.
Historical Simulation Approach :Basic historical simulation approach.
Adjusted weighting historical simulation approach.
Volatility Adjusted approach ( EWMA & GARCH Models).
Model-Building Approach :
Equal-WeightsEWMA
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METHODOLOGY OF WORK
First of all, the normality of stocks returns was tested,
to ensure returns are normally distributed, so that no
out layer number that may effect our estimation of VaR
.
Secondly, VaR was estimated based on standardapproach.
Thirdly, under the Historical Simulation Approach, 500
alternative scenarios was built based on 501 returns of
stocks, to estimate the probability distribution of the
change in the value of the current portfolio
Finally, under the Model-Building Approach, the
covariance matrix between stocks returns was built
and used in estimation.
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NORMALITY TEST
All portfolio stocks returns are normally
distributed or semi normally.
The descriptive analysis of returns was made,
and the values of Kurtosis and Skewness was
checked as follow:
Skewness Kurtosis Stock
0.304856 8.884672 AHLI
-0.331869 0.617962 JOCM
0.591727 1.772698 JPPC
-0.189063 3.460736 JODA
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STANDARD APPROACH
Based on returns of portfolio, the
mean and standard deviation was
calculated.
The following formula was used to
estimate VaR :
)(
1XNVaR
=
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HISTORICAL SIMULATION
APPROACH
Basic Historical Simulation:According to Basic historical simulation approach:
VaR is based on historical scenarios of losses .we collect the historical data on their
returns over a set observation period Each scenario -or day outcome- is given equal
weight, which is 1/number of scenarios .
for each asset and each t in the observation period, we generate scenarios by
calculating the return (% change) on each of the assets. Here is the formula to
calculate the percentage price changes: (price t - price t-1) / price t-1 or (ln t) .
For 500 scenarios , the one-day 99% VaR can be estimated as the fifth-worst loss.
no of observations=500, 1-.99=10% ,10%*500=5 Then we find mean and standard deviation and according to the equation: Mean-Z(n)*standard deviation Wefind the historical var
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HISTORICAL SIMULATION
APPROACH
We suggest that more recent observations should be given more weights because theyare more reflective of current volatilities and current macroeconomic conditions .
We calculate the weights by choosing lambda = 0.94
and our formula .
n-i(1-)
1-n
Where
n: number of observation .
i : scenario number,
i=1 is the scenario that calculated from the most distant data.
: can be chosen by experimenting to see which value back-test best .
As approaches 1, the relative weights are approach the equal weight. Then we do a cumulative weight column for our weights
Starting at the most worst observation sum weight until the required quintile of
distribution is reached( we are calculating VaR with 99% confidence level) , so we
continue summing weight until the cumulative weight is just greater that 0.01 .
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CALCULATING VAR USING THE HISTORICAL
SIMULATION VOLATILITY-ADJUSTED
APPROACH (EWMA)
In this part volatility of each scenario
was taken into consideration .
This approach will produce VAR
estimation that incorporate the
volatility of current information
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PROCEDURE OF VOLATILITY
ESTIMATION USING EWMA
Calculate daily varianceThe following equation used to produce new variance
Then we find standard deviation which is the
Square root of variance Then we make volatility multiplier:Last sd/1stsd, last sd/2ndsd Then we multiply volatility*lossesVar=1-95%=5% we will find the 5thloss from the
bottom
assumed to be .94Adjusted prices are then calculated using the followingformula :
2
1
2
1
2 )1( += nnn u
1
111
/)(
+ +
i
iniii
n
v
vvv
v
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MODEL-BUILDING APPROACH
The main alternative to historical
simulation is to make assumptions
about the probability distributions of
the returns on the market variablesThis is known as themodel
building approach(or sometimes
the variance-covariance approach).
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Daily changes in the value of a portfolio equal the
total daily changes in the values of individual
stocks.
This approach based on the assumption that daily
changes of the values of individual stocks are
normally distributed and so daily changes in the
value of the portfolio are normally distributed.
The variance of the daily changes of portfolio value
is given by:
j
n
i
n
j
iijP = =
=1 1
2cov
MODEL-BUILDING APPROACH
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PROCEDURE OF ESTIMATING VAR
IN MODEL-BUILDING APPROACH
Calculate daily returns for each stock.
Based on the following equation, the
variance of portfolio is calculated
j
n
i
n
j
iijP = =
=1 1
2 cov
Then VaR of the portfolio is estimated.
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USING EWMA IN MODEL-
BUILDING APPROACH
Instead of using equal weights, Exponentially
weighted average method with certain value could
be used.
Firstly, calculate variance of each stocks returnsusing .
Secondly, calculate covariance for each pair of stocks
using .
Finally, build the variance-covariance matrix tocalculate portfolio variance
Then VaR of the portfolio can be estimated.